 # la) Prove that the infinite product converges normally in C. thus ...

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la) Prove that the infinite product converges normally in C. thus f(z) is an entire function. b) Relate f(z) to the function g(z)=sinz Hint: Use that 2) Consider for Res&gt;0 a) Prove that, as n 100 the partial sums converge uniformly on every compact subset K of the half-plane H={s : Res&gt;0} This holds although the series defining h(s) does not converge absolutely unless Res &gt; 1. Thus, h(s) is holomorphic in the half-plane H. b) Prove the h(s)=(1-21-s)s(s)forRes&gt;1 Hint: Consider (s) h(s). c) Prove that h(s)21-1&gt;0 for s&gt;0. d) Prove that (s) &lt;0 for 0&lt;8&lt;1 Here (s) denotes the analytic continuation of the function 1 Res&gt;1. 3) Let aj denote a sequence of complex numbers. Then (0.1) is called a Dirichlet series. Assume that Res&gt;1. Prove: If the sequence aj is bounded then the series (0.1) converges absolutely and defines a function f(s) which is holomorphic in the half-plane H={s: Res&gt;1}. 4) Let aj and bk denote two bounded sequences of complex numbers and let for Prove that for Res&gt;1 (0.2) where jk=n The sum defining Cn is the sum of all products ajbk where j and k are positive integers with jk = n. Prove that the series for that holds 1) Let Pj denote the j-th prime, i.e., p1=2,p2= 3,p3 = 5. etc. Determine the following limits: and lim Hint: Recall what we learned about 2) Let denote the straight line with parametrization s(t)=c+it,-00<t<00 Prove the formula if O<a<1 a="">1 for c&gt;0. Hint: This is a result in Chapter 7 of [Stein, Shakarchi]. Carry out the details 3) Recall the reflection formula for the T-function for 8EC\Z. Use the formula to prove for ter. 4a) Use the Prime Number Theorem to prove that x-00 In r b) Let Pj denote the j-th prime number. Prove that Hint: Use the PNT for x = p, and use a). </a<1></t<00

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